In this article, we consider the nonlinear stochastic partial differential equation of fractional order in both space and time variables with constant initial condition: \begin{ equation*} \left(\partial^{ \beta}_t+\dfrac{ \nu}{ 2}\left(-\Delta\right)^{ \alpha / 2}\right) u(t, x)= ~ I_{ t}^{ \gamma}\left[\lambda u(t, x) \dot{ W}(t, x)\right] \quad t>0,~ x\in\mathbb R^d, \end{ equation*} where $\dot{ W}$ is space-time white noise, $\alpha>0$, $\beta\in(0,2]$, $\gamma \ge 0$, $\lambda\neq0$ and $\nu>0$. The existence and uniqueness of solution in the It\^o-Skorohod sense is obtained under Dalang's condition. We obtain explicit formulas for both the second moment and the second moment Lyapunov exponent. We derive the $p$-th moment upper bounds and find the matching lower bounds. Our results solve a large class of conjectures regarding the order of the $p$-th moment Lyapunov exponents. In particular, by letting $\beta=2$, $\alpha=2$, $\gamma=0$, and $d=1$, we confirm the following standing conjecture for the stochastic wave equation: \begin{ align*} t^{ -1}\log\mathbb E[u(t,x)^p] \asymp p^{ 3/2}, \quad \text{ for $p\ge 2$ as $t\to \infty$.} \end{ align*} The method for the lower bounds is inspired by a recent work by Hu and Wang [HW21], where the authors focus on the space-time colored Gaussian noise.
Categories: math.PR